STRATIFICATIONS OF THE RAY SPACE OF A TROPICAL QUADRATIC FORM BY CAUCHY–SCHWARTZ FUNCTIONS

Classes of an equivalence relation on a module V over a supertropical semiring, called rays, carry the underlying structure of ‘supertropical trigonometry’ and thereby a version of convex geometry which is compatible with quasilinearity. In this theory, the traditional Cauchy–Schwarz inequality is replaced by the CS-ratio, which gives rise to special characteristic functions, called CS-functions. These functions partite the ray space Ray(V) into convex sets and establish the main tool for analyzing varieties of quasilinear stars in Ray(V). They provide stratifications of Ray(V) and, therefore, a finer convex analysis that helps better understand geometric properties.